Newspace parameters
Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(12.2723972812\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 13) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | 7.00000 | 0 | −7.00000 | 0 | 13.0000 | 0 | 22.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 208.4.a.g | 1 | |
3.b | odd | 2 | 1 | 1872.4.a.k | 1 | ||
4.b | odd | 2 | 1 | 13.4.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 832.4.a.a | 1 | ||
8.d | odd | 2 | 1 | 832.4.a.r | 1 | ||
12.b | even | 2 | 1 | 117.4.a.b | 1 | ||
20.d | odd | 2 | 1 | 325.4.a.d | 1 | ||
20.e | even | 4 | 2 | 325.4.b.b | 2 | ||
28.d | even | 2 | 1 | 637.4.a.a | 1 | ||
44.c | even | 2 | 1 | 1573.4.a.a | 1 | ||
52.b | odd | 2 | 1 | 169.4.a.e | 1 | ||
52.f | even | 4 | 2 | 169.4.b.a | 2 | ||
52.i | odd | 6 | 2 | 169.4.c.a | 2 | ||
52.j | odd | 6 | 2 | 169.4.c.e | 2 | ||
52.l | even | 12 | 4 | 169.4.e.e | 4 | ||
156.h | even | 2 | 1 | 1521.4.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
117.4.a.b | 1 | 12.b | even | 2 | 1 | ||
169.4.a.e | 1 | 52.b | odd | 2 | 1 | ||
169.4.b.a | 2 | 52.f | even | 4 | 2 | ||
169.4.c.a | 2 | 52.i | odd | 6 | 2 | ||
169.4.c.e | 2 | 52.j | odd | 6 | 2 | ||
169.4.e.e | 4 | 52.l | even | 12 | 4 | ||
208.4.a.g | 1 | 1.a | even | 1 | 1 | trivial | |
325.4.a.d | 1 | 20.d | odd | 2 | 1 | ||
325.4.b.b | 2 | 20.e | even | 4 | 2 | ||
637.4.a.a | 1 | 28.d | even | 2 | 1 | ||
832.4.a.a | 1 | 8.b | even | 2 | 1 | ||
832.4.a.r | 1 | 8.d | odd | 2 | 1 | ||
1521.4.a.a | 1 | 156.h | even | 2 | 1 | ||
1573.4.a.a | 1 | 44.c | even | 2 | 1 | ||
1872.4.a.k | 1 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 7 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 7 \)
$5$
\( T + 7 \)
$7$
\( T - 13 \)
$11$
\( T - 26 \)
$13$
\( T - 13 \)
$17$
\( T - 77 \)
$19$
\( T - 126 \)
$23$
\( T - 96 \)
$29$
\( T + 82 \)
$31$
\( T + 196 \)
$37$
\( T + 131 \)
$41$
\( T - 336 \)
$43$
\( T - 201 \)
$47$
\( T - 105 \)
$53$
\( T + 432 \)
$59$
\( T - 294 \)
$61$
\( T + 56 \)
$67$
\( T + 478 \)
$71$
\( T + 9 \)
$73$
\( T - 98 \)
$79$
\( T + 1304 \)
$83$
\( T - 308 \)
$89$
\( T + 1190 \)
$97$
\( T - 70 \)
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